# Coulomb Gases and Ginzburg–Landau Vortices

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*Sylvia Serfaty*

A publication of the European Mathematical Society

The topic of this book is systems of points in Coulomb interaction,
in particular, the classical Coulomb gas, and vortices in the
Ginzburg–Landau model of superconductivity. The classical
Coulomb and Log gases are classical statistical mechanics models,
which have seen important developments in the mathematical literature
due to their connection with random matrices and approximation
theory. At low temperature these systems are expected to
“crystallize” to so-called Fekete sets, which exhibit
microscopically a lattice structure. The Ginzburg–Landau model,
on the other hand, describes superconductors. In superconducting
materials subjected to an external magnetic field, densely packed
point vortices emerge, forming perfect triangular lattice patterns,
so-called Abrikosov lattices.

This book describes these two systems and explores the similarity
between them. It presents the mathematical tools developed to analyze
the interaction between the Coulomb particles or the vortices, at the
microscopic scale, and describes a “renormalized energy”
governing the point patterns. This is believed to measure the disorder
of a point configuration and to be minimized by the Abrikosov lattice
in dimension 2.

This book gives a self-contained presentation of results on the mean
field limit of the Coulomb gas system, with or without temperature, and
of the derivation of the renormalized energy. It also provides a
streamlined presentation of the similar analysis that can be performed
for the Ginzburg–Landau model, including a review of the vortex-specific
tools and the derivation of the critical fields, the mean-field limit,
and the renormalized energy.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in Log gas and the Ginzburg–Landau model of superconductivity.